7 ALGEBRAIC PROPERTIES OF CONVOLUTION 277
(14.7.3) If the sequence (u^..., un) is convolvable, then so is the sequence
(14.7.3.1) (//! * fi2 * • • • * uny = £in * #„_! * • • • * /V
For (14.1.4 and 13.7.10) we have
if and only if
f-r
j-r
and these two integrals are equal.
On the other hand, if the sequence (A, u) is convolvable, it does not
necessarily follow that (/*, A) is convolvable (Problem 2). But if G is com-
mutative this will be the case, and we shall have A * \JL = fj. * A.
In particular, it follows from the preceding results that
(14.7.4) On the set M£(G) of bounded measures on G, the law of composition
(A, p) i-> A * IJL (together with the vector space structure) defines a C-algebra
structure\ the unit element is the Dirac measure se at the neutral element e ofG.
The set M£(G) of compactly supported measures on G is a subalgebra of
Mc(G). The algebra M£(G) is commutative if and only ifG is commutative.
The fact that G is commutative if M£(G) is commutative follows from the
formula (14.6.1.2).
If G is discrete, the algebra M£(G) consists of all linear combinations
]T as8&, where as = 0 for all but a finite number of points s e G (3.16.3), and
seG
the formula (14.6.1.2) shows that
seG / \seG
This is what is called in algebra the group algebra of the group G over the
field C.f*4.(G), hence is (A * ^)-integrable, and we have